Squarefree values of multivariable polynomials

By Bjorn Poonen. 2002

109 citations recorded by the source corpus.

Given f in Z[x_1,...,x_n], we compute the density of x in Z^n such that f(x) is squarefree, assuming the abc conjecture. Given f,g in Z[x_1,...,x_n], we compute unconditionally the density of x in Z^n such that gcd(f(x),g(x))=1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f in Z[x], we estimate the size of the image of f({1,2,...,n}) in (Q^*/Q^*2) union {0}.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b5bc917eb21933f26df737e498e04266f679c4cd

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Bertini theorems over finite fields (2002, methodology): “hat the abc conjecture and another conjecture imply analogues of our main theorems for regular quasiprojective schemes over SpecZ. The abc conjecture is needed to apply a multivariable generalization [Poo02] of A. Granville’s result [Gra98] about squarefree values of polynomials. For some open quest”
On the behaviour of root numbers in families of elliptic curves (2004, unknown): “Then we face a qualitatively different situation: on the one hand, A1(P ) and A2(P ) are known to hold for all square-free P ([27], [26]); on the other hand, B1(P ) and B2(P ) are false for some non-constant, square-free P ([4]).”
The geometric sieve and the density of squarefree values of invariant polynomials (2014, background): “3), we lead up to a quantitative version of a certain uniformity estimate due to Ekedahl [22] (see also Poonen [33, 34] and Poonen–Stoll [35]).”
A construction of polynomials with squarefree discriminants (2011, background): “Granville’s conditional theorem was extended to multivariate polynomials by Poonen [15]; that result implies that under abc, ∆(P ) is squarefree with the expected probability.”